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<H1><A NAME="SECTION031000000000000000000"></A><A NAME="21976"></A><A NAME="chapqrefblas"></A>
<BR>
Quick Reference 
Guide to the BLAS
</H1>

<P>

<P>
<FONT SIZE="-1">
<B>Level 1 BLAS</B>
</FONT><PRE>
                   dim scalar vector   vector   scalars              5-element prefixes
                                                                     array

SUBROUTINE _ROTG (                                      A, B, C, S )          S, D
SUBROUTINE _ROTMG(                              D1, D2, A, B,        PARAM )  S, D
SUBROUTINE _ROT  ( N,         X, INCX, Y, INCY,               C, S )          S, D
SUBROUTINE _ROTM ( N,         X, INCX, Y, INCY,                      PARAM )  S, D
SUBROUTINE _SWAP ( N,         X, INCX, Y, INCY )                              S, D, C, Z
SUBROUTINE _SCAL ( N,  ALPHA, X, INCX )                                       S, D, C, Z, CS, ZD
SUBROUTINE _COPY ( N,         X, INCX, Y, INCY )                              S, D, C, Z
SUBROUTINE _AXPY ( N,  ALPHA, X, INCX, Y, INCY )                              S, D, C, Z
FUNCTION   _DOT  ( N,         X, INCX, Y, INCY )                              S, D, DS
FUNCTION   _DOTU ( N,         X, INCX, Y, INCY )                              C, Z
FUNCTION   _DOTC ( N,         X, INCX, Y, INCY )                              C, Z
FUNCTION   __DOT ( N,  ALPHA, X, INCX, Y, INCY )                              SDS
FUNCTION   _NRM2 ( N,         X, INCX )                                       S, D, SC, DZ
FUNCTION   _ASUM ( N,         X, INCX )                                       S, D, SC, DZ
FUNCTION   I_AMAX( N,         X, INCX )                                       S, D, C, Z
</PRE>
<P>
<FONT SIZE="-1">
<BR>
<BR>
</FONT>
<P>
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Name   </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">Operation                     </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Prefixes </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_ROTG </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Generate plane rotation 
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_ROTMG </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Generate modified plane rotation
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_ROT </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Apply plane rotation
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_ROTM </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Apply modified plane rotation
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SWAP </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftrightarrow y$
 -->
<IMG
 WIDTH="51" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img982.gif"
 ALT="$ x \leftrightarrow y $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SCAL </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftarrow \alpha x$
 -->
<IMG
 WIDTH="63" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img983.gif"
 ALT="$ x \leftarrow \alpha x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z, CS, ZD </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_COPY </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow x$
 -->
<IMG
 WIDTH="51" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img984.gif"
 ALT="$ y \leftarrow x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_AXPY </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha x + y$
 -->
<IMG
 WIDTH="92" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img207.gif"
 ALT="$y \leftarrow \alpha x + y$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_DOT </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $dot \leftarrow x ^ {T} y$
 -->
<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img985.gif"
 ALT="$ dot \leftarrow x ^ {T} y $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, DS </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_DOTU </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $dot \leftarrow x ^ {T} y$
 -->
<IMG
 WIDTH="85" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img985.gif"
 ALT="$ dot \leftarrow x ^ {T} y $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_DOTC </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $dot \leftarrow x ^ {H} y$
 -->
<IMG
 WIDTH="87" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img986.gif"
 ALT="$ dot \leftarrow x ^ {H} y $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
__DOT </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $dot \leftarrow \alpha + x ^ {T} y$
 -->
<IMG
 WIDTH="118" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img987.gif"
 ALT="$ dot \leftarrow \alpha + x ^ {T} y $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> SDS </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_NRM2 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $nrm2 \leftarrow || x || _ {2}$
 -->
<IMG
 WIDTH="111" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img988.gif"
 ALT="$ nrm2 \leftarrow \vert\vert x \vert\vert _ {2} $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, SC, DZ </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_ASUM </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $asum \leftarrow || re( x ) || _ {1}  + || im( x ) || _ {1}$
 -->
<IMG
 WIDTH="234" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img989.gif"
 ALT="$ asum \leftarrow \vert\vert re( x ) \vert\vert _ {1} + \vert\vert im( x ) \vert\vert _ {1} $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, SC, DZ </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
I_AMAX </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $amax \leftarrow 1^{st} k \ni | re( x _ {k} ) |  + | im( x _ {k} ) |$
 -->
<IMG
 WIDTH="269" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="img990.gif"
 ALT="$ amax \leftarrow 1^{st} k \ni \vert re( x _ {k} ) \vert + \vert im( x _ {k} ) \vert $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
<BR>
<BR>
<BR>
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> <code>            </code> 
<!-- MATH
 $= max( | re( x _ {i} ) |  + | im( x _ {i} ) | )$
 -->
<B>= <I>max</I>( | <I>re</I>( <I>x</I> <I>i</I> ) |  + | <I>im</I>( <I>x</I> <I>i</I> ) | )</B> </FONT></TD>
<TD></TD>
</TR>
</TABLE>
<P>
<FONT SIZE="-1">
<BR>

</FONT>
<P>
<FONT SIZE="-1"><B>Level 2 BLAS</B>
</FONT><PRE>
        options            dim   b-width scalar matrix  vector   scalar vector   prefixes

_GEMV (        TRANS,      M, N,         ALPHA, A, LDA, X, INCX, BETA,  Y, INCY ) S, D, C, Z
_GBMV (        TRANS,      M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA,  Y, INCY ) S, D, C, Z
_HEMV ( UPLO,                 N,         ALPHA, A, LDA, X, INCX, BETA,  Y, INCY ) C, Z
_HBMV ( UPLO,                 N, K,      ALPHA, A, LDA, X, INCX, BETA,  Y, INCY ) C, Z
_HPMV ( UPLO,                 N,         ALPHA, AP,     X, INCX, BETA,  Y, INCY ) C, Z
_SYMV ( UPLO,                 N,         ALPHA, A, LDA, X, INCX, BETA,  Y, INCY ) S, D
_SBMV ( UPLO,                 N, K,      ALPHA, A, LDA, X, INCX, BETA,  Y, INCY ) S, D
_SPMV ( UPLO,                 N,         ALPHA, AP,     X, INCX, BETA,  Y, INCY ) S, D
_TRMV ( UPLO, TRANS, DIAG,    N,                A, LDA, X, INCX )                 S, D, C, Z
_TBMV ( UPLO, TRANS, DIAG,    N, K,             A, LDA, X, INCX )                 S, D, C, Z
_TPMV ( UPLO, TRANS, DIAG,    N,                AP,     X, INCX )                 S, D, C, Z
_TRSV ( UPLO, TRANS, DIAG,    N,                A, LDA, X, INCX )                 S, D, C, Z
_TBSV ( UPLO, TRANS, DIAG,    N, K,             A, LDA, X, INCX )                 S, D, C, Z
_TPSV ( UPLO, TRANS, DIAG,    N,                AP,     X, INCX )                 S, D, C, Z

        options            dim   scalar vector   vector   matrix  prefixes

_GER  (                    M, N, ALPHA, X, INCX, Y, INCY, A, LDA ) S, D
_GERU (                    M, N, ALPHA, X, INCX, Y, INCY, A, LDA ) C, Z
_GERC (                    M, N, ALPHA, X, INCX, Y, INCY, A, LDA ) C, Z
_HER  ( UPLO,                 N, ALPHA, X, INCX,          A, LDA ) C, Z
_HPR  ( UPLO,                 N, ALPHA, X, INCX,          AP )     C, Z
_HER2 ( UPLO,                 N, ALPHA, X, INCX, Y, INCY, A, LDA ) C, Z
_HPR2 ( UPLO,                 N, ALPHA, X, INCX, Y, INCY, AP )     C, Z
_SYR  ( UPLO,                 N, ALPHA, X, INCX,          A, LDA ) S, D
_SPR  ( UPLO,                 N, ALPHA, X, INCX,          AP )     S, D
_SYR2 ( UPLO,                 N, ALPHA, X, INCX, Y, INCY, A, LDA ) S, D
_SPR2 ( UPLO,                 N, ALPHA, X, INCX, Y, INCY, AP )     S, D
</PRE>
<P>
<FONT SIZE="-1"><B>Level 3 BLAS</B>
</FONT><PRE>
        options                          dim      scalar matrix  matrix  scalar matrix  prefixes

_GEMM (             TRANSA, TRANSB,      M, N, K, ALPHA, A, LDA, B, LDB, BETA,  C, LDC ) S, D, C, Z
_SYMM ( SIDE, UPLO,                      M, N,    ALPHA, A, LDA, B, LDB, BETA,  C, LDC ) S, D, C, Z
_HEMM ( SIDE, UPLO,                      M, N,    ALPHA, A, LDA, B, LDB, BETA,  C, LDC ) C, Z
_SYRK (       UPLO, TRANS,                  N, K, ALPHA, A, LDA,         BETA,  C, LDC ) S, D, C, Z
_HERK (       UPLO, TRANS,                  N, K, ALPHA, A, LDA,         BETA,  C, LDC ) C, Z
_SYR2K(       UPLO, TRANS,                  N, K, ALPHA, A, LDA, B, LDB, BETA,  C, LDC ) S, D, C, Z
_HER2K(       UPLO, TRANS,                  N, K, ALPHA, A, LDA, B, LDB, BETA,  C, LDC ) C, Z
_TRMM ( SIDE, UPLO, TRANSA,        DIAG, M, N,    ALPHA, A, LDA, B, LDB )                S, D, C, Z
_TRSM ( SIDE, UPLO, TRANSA,        DIAG, M, N,    ALPHA, A, LDA, B, LDB )                S, D, C, Z
</PRE>
<P>
<FONT SIZE="-1">
<BR>

</FONT>
<P>
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Name   </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">Operation                     </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Prefixes </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_GEMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y , y \leftarrow \alpha A ^ {T} x + \beta y , y \leftarrow \alpha A ^{H} x + \beta y , A - m \times n$
 -->
<IMG
 WIDTH="468" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img991.gif"
 ALT="$ y \leftarrow \alpha A x + \beta y , y \leftarrow \alpha A ^ {T} x + \beta y , y \leftarrow \alpha A ^{H} x + \beta y , A - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_GBMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y , y \leftarrow \alpha A ^ {T} x + \beta y , y \leftarrow \alpha A ^{H} x + \beta y , A - m \times n$
 -->
<IMG
 WIDTH="468" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img991.gif"
 ALT="$ y \leftarrow \alpha A x + \beta y , y \leftarrow \alpha A ^ {T} x + \beta y , y \leftarrow \alpha A ^{H} x + \beta y , A - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HEMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y$
 -->
<IMG
 WIDTH="116" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img208.gif"
 ALT="$y \leftarrow \alpha A x + \beta y$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HBMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y$
 -->
<IMG
 WIDTH="116" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img208.gif"
 ALT="$y \leftarrow \alpha A x + \beta y$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HPMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y$
 -->
<IMG
 WIDTH="116" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img208.gif"
 ALT="$y \leftarrow \alpha A x + \beta y$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SYMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y$
 -->
<IMG
 WIDTH="116" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img208.gif"
 ALT="$y \leftarrow \alpha A x + \beta y$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SBMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y$
 -->
<IMG
 WIDTH="116" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img208.gif"
 ALT="$y \leftarrow \alpha A x + \beta y$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SPMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $y \leftarrow \alpha A x + \beta y$
 -->
<IMG
 WIDTH="116" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img208.gif"
 ALT="$y \leftarrow \alpha A x + \beta y$">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_TRMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftarrow A x, x \leftarrow A ^{T} x, x \leftarrow A ^ {H} x$
 -->
<IMG
 WIDTH="224" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img992.gif"
 ALT="$ x \leftarrow A x, x \leftarrow A ^{T} x, x \leftarrow A ^ {H} x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_TBMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftarrow A x, x \leftarrow A ^{T} x, x \leftarrow A ^ {H} x$
 -->
<IMG
 WIDTH="224" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img992.gif"
 ALT="$ x \leftarrow A x, x \leftarrow A ^{T} x, x \leftarrow A ^ {H} x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_TPMV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftarrow A x, x \leftarrow A ^{T} x, x \leftarrow A ^ {H} x$
 -->
<IMG
 WIDTH="224" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img992.gif"
 ALT="$ x \leftarrow A x, x \leftarrow A ^{T} x, x \leftarrow A ^ {H} x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_TRSV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftarrow A ^{-1} x, x \leftarrow A ^{-T} x, x \leftarrow A ^ {-H} x$
 -->
<IMG
 WIDTH="263" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img993.gif"
 ALT="$ x \leftarrow A ^{-1} x, x \leftarrow A ^{-T} x, x \leftarrow A ^ {-H} x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_TBSV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftarrow A ^{-1} x, x \leftarrow A ^{-T} x, x \leftarrow A ^ {-H} x$
 -->
<IMG
 WIDTH="263" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img993.gif"
 ALT="$ x \leftarrow A ^{-1} x, x \leftarrow A ^{-T} x, x \leftarrow A ^ {-H} x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
<BR>
<BR>
<BR>
_TPSV </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $x \leftarrow A ^{-1} x, x \leftarrow A ^{-T} x, x \leftarrow A ^ {-H} x$
 -->
<IMG
 WIDTH="263" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img993.gif"
 ALT="$ x \leftarrow A ^{-1} x, x \leftarrow A ^{-T} x, x \leftarrow A ^ {-H} x $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">_GER </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x y ^{T} + A , A - m \times n$
 -->
<IMG
 WIDTH="209" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img994.gif"
 ALT="$ A \leftarrow \alpha x y ^{T} + A , A - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_GERU </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x y ^{T} + A , A - m \times n$
 -->
<IMG
 WIDTH="209" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img994.gif"
 ALT="$ A \leftarrow \alpha x y ^{T} + A , A - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_GERC </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x y ^{H} + A , A - m \times n$
 -->
<IMG
 WIDTH="211" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img995.gif"
 ALT="$ A \leftarrow \alpha x y ^{H} + A , A - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HER </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x x ^{H} + A$
 -->
<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img996.gif"
 ALT="$ A \leftarrow \alpha x x ^{H} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HPR </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x x ^{H} + A$
 -->
<IMG
 WIDTH="123" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img996.gif"
 ALT="$ A \leftarrow \alpha x x ^{H} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HER2 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x y ^{H} + y ( \alpha x ) ^ {H} + A$
 -->
<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img997.gif"
 ALT="$ A \leftarrow \alpha x y ^{H} + y ( \alpha x ) ^ {H} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HPR2 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x y ^{H} + y ( \alpha x ) ^ {H} + A$
 -->
<IMG
 WIDTH="200" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img997.gif"
 ALT="$ A \leftarrow \alpha x y ^{H} + y ( \alpha x ) ^ {H} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SYR </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x x ^{T} + A$
 -->
<IMG
 WIDTH="121" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img998.gif"
 ALT="$ A \leftarrow \alpha x x ^{T} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SPR </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x x ^{T} + A$
 -->
<IMG
 WIDTH="121" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img998.gif"
 ALT="$ A \leftarrow \alpha x x ^{T} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SYR2 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x y ^{T} + \alpha y x ^ {T} + A$
 -->
<IMG
 WIDTH="182" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img999.gif"
 ALT="$ A \leftarrow \alpha x y ^{T} + \alpha y x ^ {T} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
<BR>
<BR>
<BR>
<BR>
_SPR2 </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $A \leftarrow \alpha x y ^{T} + \alpha y x ^ {T} + A$
 -->
<IMG
 WIDTH="182" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img999.gif"
 ALT="$ A \leftarrow \alpha x y ^{T} + \alpha y x ^ {T} + A $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D </FONT></TD>
</TR>
</TABLE>
<P>
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
Name   </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">Operation                     </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> Prefixes </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_GEMM </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $C \leftarrow \alpha op(A)op(B) + \beta C, op(X) = X, X ^{T}, X ^{H}, C - m \times n$
 -->
<IMG
 WIDTH="443" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1000.gif"
 ALT="$ C \leftarrow \alpha op(A)op(B) + \beta C, op(X) = X, X ^{T}, X ^{H}, C - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SYMM </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $C \leftarrow \alpha AB + \beta C, C \leftarrow \alpha BA + \beta C, C - m \times n, A = A ^{T}$
 -->
<IMG
 WIDTH="418" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1001.gif"
 ALT="$ C \leftarrow \alpha AB + \beta C, C \leftarrow \alpha BA + \beta C, C - m \times n, A = A ^{T} $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HEMM </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $C \leftarrow \alpha AB + \beta C, C \leftarrow \alpha BA + \beta C, C - m \times n, A = A ^{H}$
 -->
<IMG
 WIDTH="420" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1002.gif"
 ALT="$ C \leftarrow \alpha AB + \beta C, C \leftarrow \alpha BA + \beta C, C - m \times n, A = A ^{H} $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SYRK </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $C \leftarrow \alpha AA ^{T} + \beta C, C \leftarrow \alpha A ^{T} A + \beta C, C - n \times n$
 -->
<IMG
 WIDTH="364" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1003.gif"
 ALT="$ C \leftarrow \alpha AA ^{T} + \beta C, C \leftarrow \alpha A ^{T} A + \beta C, C - n \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HERK </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $C \leftarrow \alpha AA ^{H} + \beta C, C \leftarrow \alpha A ^{H} A + \beta C, C - n \times n$
 -->
<IMG
 WIDTH="369" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1004.gif"
 ALT="$ C \leftarrow \alpha AA ^{H} + \beta C, C \leftarrow \alpha A ^{H} A + \beta C, C - n \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_SYR2K </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $C \leftarrow \alpha AB ^{T} + \alpha BA ^{T} + \beta C, C \leftarrow \alpha A ^{T} B + \alpha B ^{T} A + \beta C, C - n \times n$
 -->
<IMG
 WIDTH="507" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1005.gif"
 ALT="$ C \leftarrow \alpha AB ^{T} + \alpha BA ^{T} + \beta C, C \leftarrow \alpha A ^{T} B + \alpha B ^{T} A + \beta C, C - n \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_HER2K </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $C \leftarrow \alpha AB ^{H} + \bar{\alpha} BA ^{H} + \beta C, C \leftarrow \alpha A ^{H} B + \bar{\alpha} B ^{H} A + \beta C, C - n \times n$
 -->
<IMG
 WIDTH="516" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1006.gif"
 ALT="$ C \leftarrow \alpha AB ^{H} + \bar{\alpha} BA ^{H} + \beta C, C \leftarrow \alpha A ^{H} B + \bar{\alpha} B ^{H} A + \beta C, C - n \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_TRMM </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $B \leftarrow \alpha op(A)B, B \leftarrow \alpha B op(A), op(A) = A, A ^{T}, A ^{H}, B - m \times n$
 -->
<IMG
 WIDTH="476" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1007.gif"
 ALT="$ B \leftarrow \alpha op(A)B, B \leftarrow \alpha B op(A), op(A) = A, A ^{T}, A ^{H}, B - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
_TRSM </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> 
<!-- MATH
 $B \leftarrow \alpha op(A ^{-1} )B, B \leftarrow \alpha B op(A ^{-1} ), op(A) = A, A ^{T}, A ^{H}, B - m \times n$
 -->
<IMG
 WIDTH="512" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img1008.gif"
 ALT="$ B \leftarrow \alpha op(A ^{-1} )B, B \leftarrow \alpha B op(A ^{-1} ), op(A) = A, A ^{T}, A ^{H}, B - m \times n $">
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> S, D, C, Z </FONT></TD>
</TR>
<TR><TD></TD>
<TD></TD>
<TD></TD>
</TR>
</TABLE>
<P>
<FONT SIZE="-1">
<BR>

</FONT>
<P>

<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>
<BR>
<B>Notes</B>
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>
</FONT>
<P>
<FONT SIZE="-1"><B>Meaning of prefixes</B>
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>
</FONT>
<P>
<TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
S - REAL              </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  C - COMPLEX </FONT></TD>
<TD></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
D - DOUBLE PRECISION  </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  Z - COMPLEX*16    </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> (this may not be supported </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
</FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">                    </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1"> by all machines) </FONT></TD>
</TR>
</TABLE>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>

</FONT>
<P>
<FONT SIZE="-1">For the Level 2 BLAS a set of extended-precision routines with the prefixes
ES, ED, EC, EZ may also be available.
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>
</FONT>
<P>
<FONT SIZE="-1"><B>Level 1 BLAS </B>
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>

</FONT>
<P>
<FONT SIZE="-1">In addition to the listed routines there are two further 
extended-precision dot product routines DQDOTI and DQDOTA.
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>
</FONT>
<P>
<FONT SIZE="-1"><B>Level 2 and Level 3 BLAS </B>
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>
</FONT>
<P>
<FONT SIZE="-1"><I>Matrix types</I>
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>
</FONT>
<P>
<TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
GE - GEneral     </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  GB - General Band </FONT></TD>
<TD></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
SY - SYmmetric   </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  SB - Symmetric Band    </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  SP - Symmetric Packed </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
HE - HErmitian   </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  HB - Hermitian Band    </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  HP - Hermitian Packed </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
TR - TRiangular  </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  TB - Triangular Band   </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  TP - Triangular Packed </FONT></TD>
</TR>
</TABLE>
<P>
<FONT SIZE="-1"><I>Options</I>
</FONT>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>

</FONT>
<P>
<FONT SIZE="-1">Arguments describing options are declared as CHARACTER*1 and may be passed as character strings.
</FONT>
<P>
<TABLE CELLPADDING=3>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1">
TRANS  </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  = `</FONT><FONT SIZE="-1"><B> N</B>o transpose',  `</FONT><FONT SIZE="-1"><B> T</B>ranspose',  `</FONT><FONT SIZE="-1"><B> C</B>onjugate transpose' 
<!-- MATH
 $( X, X ^{T}, X^{C} )$
 -->
<B>( <I>X</I>, <I>X</I> <SUP><I>T</I></SUP>, <I>X</I><SUP><I>C</I></SUP> )</B> </FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
UPLO  </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  = `</FONT><FONT SIZE="-1"><B> U</B>pper triangular',  `</FONT><FONT SIZE="-1"><B> L</B>ower triangular'</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
DIAG  </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  = `</FONT><FONT SIZE="-1"><B> N</B>on-unit triangular',  `</FONT><FONT SIZE="-1"><B> U</B>nit triangular'</FONT></TD>
</TR>
<TR><TD ALIGN="LEFT"><FONT SIZE="-1"> 
SIDE  </FONT></TD>
<TD ALIGN="LEFT"><FONT SIZE="-1">  = `</FONT><FONT SIZE="-1"><B> L</B>eft',  `</FONT><FONT SIZE="-1"><B> R</B>ight' (A or op(A) on the left, or A or op(A)
on the right) </FONT></TD>
</TR>
</TABLE>
<P>
<FONT SIZE="-1">
<BR>
<BR>
<BR>

</FONT>
<DIV ALIGN="LEFT"><FONT SIZE="-1">
For real matrices, TRANS = `T' and TRANS = `C' have the same meaning.
<BR>
For Hermitian matrices, TRANS = `T' is not allowed.
<BR>
For complex symmetric matrices, TRANS = `H' is not allowed.
<BR>
</FONT></DIV>
<P>

<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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